Problem: Simplify the following expression: $ x = \dfrac{-1}{6} + \dfrac{r - 9}{-2r + 3} $
Solution: In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-2r + 3}{-2r + 3}$ $ \dfrac{-1}{6} \times \dfrac{-2r + 3}{-2r + 3} = \dfrac{2r - 3}{-12r + 18} $ Multiply the second expression by $\dfrac{6}{6}$ $ \dfrac{r - 9}{-2r + 3} \times \dfrac{6}{6} = \dfrac{6r - 54}{-12r + 18} $ Therefore $ x = \dfrac{2r - 3}{-12r + 18} + \dfrac{6r - 54}{-12r + 18} $ Now the expressions have the same denominator we can simply add the numerators: $x = \dfrac{2r - 3 + 6r - 54}{-12r + 18} $ $x = \dfrac{8r - 57}{-12r + 18}$ Simplify the expression by dividing the numerator and denominator by -1: $x = \dfrac{-8r + 57}{12r - 18}$